scholarly journals Multiwinner Voting with Restricted Admissible Sets: Complexity and Strategyproofness

2018 ◽  
Author(s):  
Yongjie Yang ◽  
Jianxin Wang
Keyword(s):  
Special Class ◽  
Voting Rules ◽  
Induced Subgraphs ◽  
Selection Rules ◽  
Admissible Sets ◽  
Graph Classes ◽  
Vertex Set ◽  
Candidate Set

Multiwinner voting aims to select a subset of candidates (the winners) from admissible sets, according to the votes cast by voters. A special class of multiwinner rules—the k-committee selection rules where the number of winners is predefined—have gained considerable attention recently. In this setting, the admissible sets are all subsets of candidates of size exactly k. In this paper, we study admissible sets with combinatorial restrictions. In particular, in our setting, we are given a graph G whose vertex set is the candidate set. Admissible sets are the subsets of candidates whose induced subgraphs belong to some special class G of graphs. We consider different graph classes G and investigate the complexity of multiwinner determination problem for prevalent voting rules in this setting. In addition, we investigate the strategyproofness of many rules for different classes of admissible sets.

scholarly journals Tree Cover Number and Maximum Semidefinite Nullity of Some Graph Classes

2020 ◽  
Vol 36 (36) ◽  
pp. 678-693
Author(s):  
Rachel Domagalski ◽  
Sivaram Narayan
Keyword(s):  
Line Graph ◽  
Cartesian Product ◽  
Positive Semidefinite ◽  
Tree Cover ◽  
Induced Subgraphs ◽  
Graph Classes ◽  
Maximum Nullity ◽  
Minimum Number ◽  
Vertex Set ◽  
Vertex Disjoint

Let $G$ be a graph with a vertex set $V$ and an edge set $E$ consisting of unordered pairs of vertices. The tree cover number of $G$, denoted $\tau(G)$, is the minimum number of vertex disjoint simple trees occurring as induced subgraphs of $G$ that cover all the vertices of $G$. In this paper, the tree cover number of a line graph $\tau(L(G))$ is shown to be equal to the path number $\pi(G)$ of $G$. Also, the tree cover numbers of shadow graphs, corona and Cartesian product of two graphs are found. The graph parameter $\tau(G)$ is related to another graph parameter $M_+(G)$, called the maximum semidefinite nullity of $G$. Suppose $S_+(G,\mathbb{R})$ denotes the collection of positive semidefinite real symmetric matrices associated with a given graph $G$. Then $M_+(G)$ is the maximum nullity among all matrices in $S_+(G,\mathbb{R})$. It has been conjectured that $\tau(G)\leq M_+(G)$. The conjecture is shown to be true for graph classes considered in this work.

INDUCED SUBGRAPHS OF GAMMA GRAPHS

2013 ◽  
Vol 05 (03) ◽  
pp. 1350012 ◽  
Author(s):  
N. SRIDHARAN ◽  
S. AMUTHA ◽  
S. B. RAO
Keyword(s):  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Vertex Set ◽  
Isomorphic Graphs

Let G be a graph. The gamma graph of G denoted by γ ⋅ G is the graph with vertex set V(γ ⋅ G) as the set of all γ-sets of G and two vertices D and S of γ ⋅ G are adjacent if and only if |D ∩ S| = γ(G) – 1. A graph H is said to be a γ-graph if there exists a graph G such that γ ⋅ G is isomorphic to H. In this paper, we show that every induced subgraph of a γ-graph is also a γ-graph. Furthermore, if we prove that H is a γ-graph, then there exists a sequence {Gn} of non-isomorphic graphs such that H = γ ⋅ Gn for every n.

scholarly journals Graphs and complete intersection toric ideals

2015 ◽  
Vol 14 (09) ◽  
pp. 1540011 ◽  
Author(s):  
I. Bermejo ◽  
I. García-Marco ◽  
E. Reyes
Keyword(s):  
Complete Intersection ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Point Of View ◽  
Toric Ideals ◽  
Induced Subgraphs ◽  
Toric Ideal ◽  
Vertex Set ◽  
Combinatorial Point ◽  
Ring Graph

Our purpose is to study the family of simple undirected graphs whose toric ideal is a complete intersection from both an algorithmic and a combinatorial point of view. We obtain a polynomial time algorithm that, given a graph G, checks whether its toric ideal PG is a complete intersection or not. Whenever PG is a complete intersection, the algorithm also returns a minimal set of generators of PG. Moreover, we prove that if G is a connected graph and PG is a complete intersection, then there exist two induced subgraphs R and C of G such that the vertex set V(G) of G is the disjoint union of V(R) and V(C), where R is a bipartite ring graph and C is either the empty graph, an odd primitive cycle, or consists of two odd primitive cycles properly connected. Finally, if R is 2-connected and C is connected, we list the families of graphs whose toric ideals are complete intersection.

scholarly journals Smaller Extended Formulations for Spanning Tree Polytopes in Minor-closed Classes and Beyond

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Manuel Aprile ◽  
Samuel Fiorini ◽  
Tony Huynh ◽  
Gwenaël Joret ◽  
David R. Wood
Keyword(s):  
Spanning Tree ◽  
Short Proof ◽  
Induced Subgraphs ◽  
Direct Construction ◽  
Extension Complexity ◽  
Graph Classes ◽  
Closed Class ◽  
Vertex Graph ◽  
Graph Class ◽  
Strongly Sublinear

Let $G$ be a connected $n$-vertex graph in a proper minor-closed class $\mathcal G$. We prove that the extension complexity of the spanning tree polytope of $G$ is $O(n^{3/2})$. This improves on the $O(n^2)$ bounds following from the work of Wong (1980) and Martin (1991). It also extends a result of Fiorini, Huynh, Joret, and Pashkovich (2017), who obtained a $O(n^{3/2})$ bound for graphs embedded in a fixed surface. Our proof works more generally for all graph classes admitting strongly sublinear balanced separators: We prove that for every constant $\beta$ with $0<\beta<1$, if $\mathcal G$ is a graph class closed under induced subgraphs such that all $n$-vertex graphs in $\mathcal G$ have balanced separators of size $O(n^\beta)$, then the extension complexity of the spanning tree polytope of every connected $n$-vertex graph in $\mathcal{G}$ is $O(n^{1+\beta})$. We in fact give two proofs of this result, one is a direct construction of the extended formulation, the other is via communication protocols. Using the latter approach we also give a short proof of the $O(n)$ bound for planar graphs due to Williams (2002).

scholarly journals Convex Partitions of Graphs induced by Paths of Order Three

2010 ◽  
Vol Vol. 12 no. 5 (Graph and Algorithms) ◽  
Author(s):  
C. C. Centeno ◽  
S. Dantas ◽  
M. C. Dourado ◽  
Dieter Rautenbach ◽  
Jayme Luiz Szwarcfiter
Keyword(s):  
Convex Sets ◽  
Graph Classes ◽  
Convex Partitions ◽  
Vertex Set ◽  
International Audience ◽  
Np Complete

Graphs and Algorithms International audience A set C of vertices of a graph G is P(3)-convex if v is an element of C for every path uvw in G with u, w is an element of C. We prove that it is NP-complete to decide for a given graph G and a given integer p whether the vertex set of G can be partitioned into p non-empty disjoint P(3)-convex sets. Furthermore, we study such partitions for a variety of graph classes.

scholarly journals A Note on the Speed of Hereditary Graph Properties

10.37236/644 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Vadim V. Lozin ◽  
Colin Mayhill ◽  
Victor Zamaraev
Keyword(s):  
Practical Importance ◽  
Bipartite Graphs ◽  
Hereditary Property ◽  
Induced Subgraphs ◽  
Forbidden Induced Subgraphs ◽  
Graph Properties ◽  
Graph Property ◽  
Vertex Set ◽  
Split Graphs ◽  
Hereditary Properties

For a graph property $X$, let $X_n$ be the number of graphs with vertex set $\{1,\ldots,n\}$ having property $X$, also known as the speed of $X$. A property $X$ is called factorial if $X$ is hereditary (i.e. closed under taking induced subgraphs) and $n^{c_1n}\le X_n\le n^{c_2n}$ for some positive constants $c_1$ and $c_2$. Hereditary properties with the speed slower than factorial are surprisingly well structured. The situation with factorial properties is more complicated and less explored, although this family includes many properties of theoretical or practical importance, such as planar graphs or graphs of bounded vertex degree. To simplify the study of factorial properties, we propose the following conjecture: the speed of a hereditary property $X$ is factorial if and only if the fastest of the following three properties is factorial: bipartite graphs in $X$, co-bipartite graphs in $X$ and split graphs in $X$. In this note, we verify the conjecture for hereditary properties defined by forbidden induced subgraphs with at most 4 vertices.

scholarly journals Forbidden Subgraphs of Power Graphs

10.37236/9961 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Pallabi Manna ◽  
Peter J. Cameron ◽  
Ranjit Mehatari
Keyword(s):  
Nilpotent Groups ◽  
Chordal Graphs ◽  
Forbidden Subgraphs ◽  
Induced Subgraphs ◽  
Open Problems ◽  
Split Graph ◽  
Forbidden Induced Subgraphs ◽  
Power Graph ◽  
Graph Classes ◽  
Split Graphs

The undirected power graph (or simply power graph) of a group $G$, denoted by $P(G)$, is a graph whose vertices are the elements of the group $G$, in which two vertices $u$ and $v$ are connected by an edge between if and only if either $u=v^i$ or $v=u^j$ for some $i$, $j$. A number of important graph classes, including perfect graphs, cographs, chordal graphs, split graphs, and threshold graphs, can be defined either structurally or in terms of forbidden induced subgraphs. We examine each of these five classes and attempt to determine for which groups $G$ the power graph $P(G)$ lies in the class under consideration. We give complete results in the case of nilpotent groups, and partial results in greater generality. In particular, the power graph is always perfect; and we determine completely the groups whose power graph is a threshold or split graph (the answer is the same for both classes). We give a number of open problems.

scholarly journals Perfect Codes Over Induced Subgraphs of Unit Graphs of Ring of Integers Modulo n

2021 ◽  
Vol 20 ◽  
pp. 399-403
Author(s):  
Mohammad Hassan Mudaber ◽  
Nor Haniza Sarmin ◽  
Ibrahim Gambo
Keyword(s):  
Perfect Code ◽  
Perfect Codes ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Ring Of Integers ◽  
Unit Graph ◽  
Vertex Set

The induced subgraph of a unit graph with vertex set as the idempotent elements of a ring R is a graph which is obtained by deleting all non idempotent elements of R. Let C be a subset of the vertex set in a graph Γ. Then C is called a perfect code if for any x, y ∈ C the union of the closed neighbourhoods of x and y gives the the vertex set and the intersection of the closed neighbourhoods of x and y gives the empty set. In this paper, the perfect codes in induced subgraphs of the unit graphs associated with the ring of integer modulo n, Zn that has the vertex set as idempotent elements of Zn are determined. The rings of integer modulo n are classified according to their induced subgraphs of the unit graphs that accept a subset of a ring Zn of different sizes as the perfect codes

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